Ising antiferromagnet on tetrahedron cactus lattices with randomness in the exchange interactions is studied. Instability line of the paramagnetic state, beyond which glassy or antiferromagnetic state is (meta)stable is obtained. The model is investigated by the replica method. Instability toward antiferromagnetic state does not occur for M <= 4 where M is the number of corner sharing tetrahedra for the cactus lattices. Instability toward glassy state occurs at as weak randomness as J/(-J(0)) similar or equal to 0.056, 0.020, and <= 10(-4) for M = 2, 3, and 4, respectively, where J(0) and J(2) are the mean and variance of the Gaussian random exchange interaction, respectively. (C) 2017 Elsevier B.V. All rights reserved.